When Is a Straight Line Not the Shortest Distance between Two Points?

In the Pythagorean theorem episode of NBC Learn's "The Science of NFL Football," you see that a defender in the middle of the field must take the proper angle of pursuit to catch a ball-carrier making a dash down the sideline for the end zone.

In chasing down the ball-carrier, the defender basically runs along the diagonal of a right triangle, in which the sum of the squares of the sides equals the square of that diagonal. You might know this relation, discovered by the 5th-century B.C. Greek mathematician Pythagoras, as a² + b² = c².

The "c " is the hypotenuse, and although it represents the longest side of a right triangle, it is the shortest path between the two points on either end. If the points on the triangle were places to visit in a city, you probably wouldn't bother walking along a and b if you could directly take c.

But the hypotenuse isn't always the shortest route. In fact, it is only the shortest one on football fields and other flat surfaces. On spheres and other shapes, it may not be.

You can see this distinction if you draw a right triangle on a globe. First, let's pick a city on the equator--for simplicity, say it's Quito, Ecuador, on the Pacific coast of South America. From Quito, trace a longitude line to the north pole; then make a 90-degree turn to the right and head straight back down. At the equator, you'll notice a city nearby called Libreville, the capital of the country of Gabon in Africa.

Now draw a line along the globe's surface starting at Quito and going toward Libreville. You probably went eastward, passing over Brazil and the Atlantic Ocean. Indeed, this hypotenuse, traversing one quarter of the globe, marks the shortest distance. But that is not the only hypotenuse.

Mathematically speaking, you would still have a right triangle if you went westward from Quito, circumnavigating Earth along the equator to get to Libreville. The hypotenuse in this case is three fourths of the circumference. It would have been shorter to travel from Quito to the North Pole and then down to Libreville.

The Pythagorean theorem only works on two-dimensional surfaces like football fields; mathematicians refer to such surfaces as Euclidean geometry (named for Euclid, the 3rd-century B.C. Greek mathematician). The theorem fails for non-Euclidean geometries, such as spheres and more complex geometries like saddles. Indeed, all the rules you learned in school, like parallel lines staying parallel, only refer to Euclidean geometry. In the non-Euclidean universe, parallel lines may actually diverge or converge.

Although non-Euclidean geometry may seem exotic and unfamiliar, it is actually common in many fields of science--perhaps most notably, in Einstein's general relativity theory, in which gravity can bend the shape of space and time.